ch04 1 s1 - michigan state university€¦ · 4.1 the concepts of force and mass a force is a push...

Chapter 4

Forces and Newton’s Laws of Motion

Prof. Bromberg will miss Wed. Office hrs. Please email for appt. on Thurs-Fri

4.1 The Concepts of Force and Mass

A force is a push or a pull acting on an object. A force is a vector!

Contact forces arise from physical contact, and are due to a stretch or compression at the point of contact.

Action-at-a-distance forces do not require contact and include gravity and forces due to charged particles



Arrows are used to represent force vectors. The length of the arrow is proportional to the magnitude of the force.

15 N = fifteen newtons

5 N force 5 N ~ 1 lb



Bold letter with arrow is the symbol, , for a force vector: has magnitude and direction.

Direction is given as an angle, , or coded in components, .

F

F

Fyθ

θ

Fx

⎧

⎨

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

Fx , Fy

F = |F |

θ = tan−1(Fy Fx )

F = Fx2 + Fy

2

Fx = F cosθFy = F sinθ



Mass of an object is a measure of the number and type of atoms within the object. Mass can be measured without resorting to gravity/weight.

A spring will oscillate a mass with an oscillation period,

T ∝ m. ( ∝ means proportional to)If the period is twice as long, the mass is 4 times bigger.


a planet or moon or a big spaceship (air-track unnecessary)

air-track

cart stretched spring stretched spring

These springs can be taken anywhere in the universe and used to measure the mass of any cart. Also, the stretching of these springs can be used to define the unit of force.

Device to measure a mass anywhere in the universe

SI Unit of Mass: kilogram (kg) Prof. Bromberg will miss Wed. Office hrs. Please email for appt. on Thurs-Fri

4.2 Newton’s Laws of Motion (First Law)

An object continues in a state of rest or in a state of motion at a constant speed along a straight line, unless compelled to change that state by a net force.

The net force is the vector sum of all of the forces acting on an object.

Newton’s First Law


Fi

i=1

N

∑ =F1 +F2 +F3 ++

FN

Mathematically, the net force is written as

where the Greek letter sigma denotes the vector sum of all forces acting on an object.

ONE object!

Net Force acting on ONE object


The net force on an object is the vector sum of all forces acting on that object.

The SI unit of force is the Newton (N).

Individual Forces Net Force

10 N 4 N 6 N

Top views



Individual Forces Net Force

3 N

4 N

5 N

36.7

θ is an angle with respect to x-axis

tanθ =Fy

Fx

⇒ θ = tan−1 Fy

Fx

⎛

⎝⎜

⎞

⎠⎟

θ = tan−1 34

⎛⎝⎜

⎞⎠⎟= 36.7°

No Gravity

No friction Top view Top view



Net Force

5 N 36.7

Why not

5 N

or this or this or this

5 N

Why does the picture show the force vector attached to a corner ?

5 N 5 N

Best would be this, attached to the center of object.

You will see this in most textbooks.

No Gravity

No friction Top view

Top view Top view Top view Top view



5 N 36.7 5 N

36.7

Both drawings lead to the same linear motion of the object

The object will not maintain a constant speed & direction.

The object will accelerate in this direction:

velocity

No Gravity

No friction

Top view

a



Newton’s 1st law: for an object to remain at rest, or move with constant speed & direction, the Net Force acting on it must be ZERO.

Newton’s 1st law: if the Net Force acting on a object is NOT ZERO, the velocity (magnitude, or direction, or both) must change.

So



Newton’s 1st law is often called the law of inertia.

Inertia is the natural tendency of an object to remain at rest or in motion at a constant speed along a straight line.

The mass of an object is a quantitative measure of inertia.



An inertial reference frame is one in which Newton’s law of inertia is valid.

All accelerating reference frames are non-inertial.



Warning: Newton’s 1st law can appear to be violated if you don’t recognize the existence of contact forces.

Newton’s 1st law: for an object to remain at rest, or move with constant speed & direction, the Net Force acting on it must be ZERO.



Examples (4 clicker questions): A mass hanging from a string. A mass at rest on a table. A mass at rest on a ramp. A mass sliding on a table.



A mass hanging from a string. Gravity applies a 100 N gravitational force to the object. What force component does the string apply to the object?

Clicker Question 4.1

a) string Fy = −100 N

b) string Fy = +100 N

c) string Fy = 0 N

d) string Fy = 1 N

e) A string can't make a force

+ y

gravity Fy = −100 N

String


A mass resting on a table. Gravity applies a 100 N gravitational force to the object. What force does the table apply to the object?


a) table Fy = −100 N

b) table Fy = +100 N

c) table Fy = 0 N

d) table Fy = 1 N

e) A table can't make a force

At rest: Net force must be zero

gravity Fy = −100 N

Table.


Gravity applies a 100 N gravitational force to an object at rest on a 15° ramp.

A mass at rest on a ramp.

Ramp

down the ramp component of gravity

Component of gravity pulls the mass down the the ramp

perpendicular to ramp component of gravity

Ramp


FG = −100 N

FG = −100 N

A mass at rest on a ramp. Gravity applies a 100 N gravitational force to an object at rest on a 15° ramp. The friction between ramp and object applies a force on the mass in what direction?


a) Frictional force >100 N, up the rampb) Frictional force = 100 N, up the rampc) Frictional force = 100 N, down the rampd) Frictional force < 100 N, up the rampe) A ramp can't make a force

FG = −100 N

At rest: Net force must be zero


Ramp

A mass sliding on a table. Gravity applies a force to a mass. It is sliding on a table with an initial velocity of +1 m/s. It slows while sliding to +0.5 m/s. To slow it the friction between them applies a force to the mass in what direction?


a) Upwardb) Downwardc) To the right (+)d) To the left (–)e) A table can't make a force.

Table.

v0 = +1 m/s


FG = −100 N

Newton’s Second Law When a net external force acts on an object of mass m, the acceleration that results is directly proportional to the net force and has a magnitude that is inversely proportional to the mass. The direction of the acceleration is the same as the direction of the net force.

Sum of forces acting on 1 object

4.2 Newton’s Laws of Motion (Second Law)

SI Unit for Force

This combination of units is called a newton (N). Note: it has the same units as ma.

1kg ⋅m/s2 = 1N


~ 5N = 1lb


A free-body-diagram is a diagram that represents the object and the forces that act on it.


The net force in this case is:

275 N + 395 N – 560 N = +110 N

and is directed along the + x axis of the coordinate system.


If the mass of the car is 1850 kg then, by Newton’s second law, the acceleration is


With the acceleration just calculated and starting at rest, how far has the car gone after 10s of pushing?


a) 0.059 mb) 10 mc) 0.59 md) 3.0 me) 0.3 m

a = 0.059 m/s2

The direction of force and acceleration vectors can be taken into account by using x and y components.

is equivalent to

&

Net Force in x-direction = m times a in x-direction

Net Force in y-direction = m times a in y-direction AND


A) If mass of the object is known, and all forces acting on the object are known, then the acceleration vector can be calculated.

How to use Newton’s 2nd law,

B) If the acceleration vector and mass of an object are known, then the Net Force acting on the object can be calculated. It may surprise you!

C) If the acceleration vector and mass of an object are known, but the calculated Net Force and the identified forces disagree, at least one additional force must act on the object. Find it!

one object


A) If mass of the object is known, and all forces acting on the object are known, then the acceleration vector can be calculated.

⇓Once the Net Force acting on an object and Newton’s 2nd law are used to calculate the object’s acceleration vector, future changes of the position and velocity can be predicted.


ax =

Fx∑m

ay =Fy∑

m

B) If the acceleration vector and mass of an object are known, then the Net Force acting on the object can be calculated.

A paddle ball travelling horizontally bounces off a wall. The speed of the ball was 30 m/s before and after hitting the wall. If contact with the wall was for 0.02 s, what was the ball’s acceleration during the contact?

v0 = +30m/s wall

v = −30m/s

If the paddle ball has a mass of 0.2 kg, what is the force that the wall applied to the ball?

one object

wall

F = ?

B) If the acceleration vector and mass of an object are known, then the Net Force acting on the object can be calculated.

A paddle ball travelling horizontally bounces off a wall. The speed of the ball was 30 m/s before and after hitting the wall. If contact with the wall was for 0.02 s, what was the ball’s acceleration during the contact?

v0 = +30m/s wall

v = −30m/s

If the paddle ball has a mass of 0.2 kg, what is the force that the wall applied to the ball?

Fx = max

= (0.2 kg)(–3000 m/s2 )= −600 kg-m/s2 or − 600 N

a =

v − v0( )t

=−30− (+30)⎡⎣ ⎤⎦m/s

.02 s= −3000 m/s2

Force on ball is to the LEFTMagnitude of 600N

(~120 lbs)

one object

wall

F = −600N

C) If the acceleration vector and mass of an object are known, but the calculated Net Force and the identified forces disagree, at least one additional force must act on the object. Likely you will not know the origin of this force, but it must be there.

A mass sliding on a table.

A 2 kg mass slides on a table with an initial velocity of +1 m/s. It slows while sliding to +0.5 m/s, in 2 seconds.

Table.

v0 = +1 m/s

1) Calculate the acceleration vector 2) Use Newton’s 2nd law, to calculate the frictional force that must act on the mass.

Table.

v0 = +1 m/s v = 0.5 m/s

m = 2 kg


a) F = −0.5Nb) F = +0.5Nc) F = −1.0Nd) F = +1.0Ne) F = −2.0N

m = 2 kg

A 2.0 kg mass sliding on a table with an initial velocity of +1.0 m/s, slows to +0.5 m/s, in 2.0 seconds.

A force acting on the mass causes it to lose speed. What is the magnitude and direction of this force?

ch04 1 s1 - michigan state university€¦ · 4.1 the concepts of force and mass a force is a push...

Documents